arXiv subjects

Pawel Pralat

Publications and source records attributed to Pawel Pralat.

At least 37 records · Page 2Linked to original sources

arXiv2023

Self-similarity of Communities of the ABCD Model

The Artificial Benchmark for Community Detection (ABCD) graph is a random graph model with community structure and power-law distribution for both degrees and community sizes. The model generates graphs similar to the well-known LFR model but it is faster and can be investigated analytically. In this paper, we show that the ABCD model exhibits some interesting self-similar behaviour, namely, the degree distribution of ground-truth communities is asymptotically the same as the degree distribution of the whole graph (appropriately normalized based on their sizes). As a result, we can not only estimate the number of edges induced by each community but also the number of self-loops and multi-edges generated during the process. Understanding these quantities is important as (a) rewiring self-loops and multi-edges to keep the graph simple is an expensive part of the algorithm, and (b) every rewiring causes the underlying configuration models to deviate slightly from uniform simple graphs on their corresponding degree sequences.

cs.SI↗

arXiv2023

Rainbow spanning trees in randomly coloured $G_{k-out}$

Given a graph $G=(V,E)$ on $n$ vertices and an assignment of colours to its edges, a set of edges $S \subseteq E$ is said to be rainbow if edges from $S$ have pairwise different colours assigned to them. In this paper, we investigate rainbow spanning trees in randomly coloured random $G_{k-out}$ graphs.

math.CO↗

arXiv2023

Unsupervised Framework for Evaluating and Explaining Structural Node Embeddings of Graphs

An embedding is a mapping from a set of nodes of a network into a real vector space. Embeddings can have various aims like capturing the underlying graph topology and structure, node-to-node relationship, or other relevant information about the graph, its subgraphs or nodes themselves. A practical challenge with using embeddings is that there are many available variants to choose from. Selecting a small set of most promising embeddings from the long list of possible options for a given task is challenging and often requires domain expertise. Embeddings can be categorized into two main types: classical embeddings and structural embeddings. Classical embeddings focus on learning both local and global proximity of nodes, while structural embeddings learn information specifically about the local structure of nodes' neighbourhood. For classical node embeddings there exists a framework which helps data scientists to identify (in an unsupervised way) a few embeddings that are worth further investigation. Unfortunately, no such framework exists for structural embeddings. In this paper we propose a framework for unsupervised ranking of structural graph embeddings. The proposed framework, apart from assigning an aggregate quality score for a structural embedding, additionally gives a data scientist insights into properties of this embedding. It produces information which predefined node features the embedding learns, how well it learns them, and which dimensions in the embedded space represent the predefined node features. Using this information the user gets a level of explainability to an otherwise complex black-box embedding algorithm.

cs.LG↗

arXiv2022

Localization Game for Random Geometric Graphs

The localization game is a two player combinatorial game played on a graph $G=(V,E)$. The cops choose a set of vertices $S_1 \subseteq V$ with $|S_1|=k$. The robber then chooses a vertex $v \in V$ whose location is hidden from the cops, but the cops learn the graph distance between the current position of the robber and the vertices in $S_1$. If this information is sufficient to locate the robber, the cops win immediately; otherwise the cops choose another set of vertices $S_2 \subseteq V$ with $|S_2|=k$, and the robber may move to a neighbouring vertex. The new distances are presented to the robber, and if the cops can deduce the new location of the robber based on all information they accumulated thus far, then they win; otherwise, a new round begins. If the robber has a strategy to avoid being captured, then she wins. The localization number is defined to be the smallest integer $k$ so that the cops win the game. In this paper we determine the localization number (up to poly-logarithmic factors) of the random geometric graph $G \in \mathcal G(n,r)$ slightly above the connectivity threshold.

math.CO↗

arXiv2022

A Fully Adaptive Strategy for Hamiltonian Cycles in the Semi-Random Graph Process

The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible. We focus on the problem of constructing a Hamiltonian cycle in as few rounds as possible. In particular, we present an adaptive strategy for the player which achieves it in $αn$ rounds, where $α< 2.01678$ is derived from the solution to some system of differential equations. We also show that the player cannot achieve the desired property in less than $βn$ rounds, where $β> 1.26575$. These results improve the previously best known bounds and, as a result, the gap between the upper and lower bounds is decreased from 1.39162 to 0.75102.

math.CO↗

arXiv2022

Modularity of the ABCD Random Graph Model with Community Structure

The Artificial Benchmark for Community Detection (ABCD) graph is a random graph model with community structure and power-law distribution for both degrees and community sizes. The model generates graphs with similar properties as the well-known LFR one, and its main parameter $ξ$ can be tuned to mimic its counterpart in the LFR model, the mixing parameter $μ$. In this paper, we investigate various theoretical asymptotic properties of the ABCD model. In particular, we analyze the modularity function, arguably, the most important graph property of networks in the context of community detection. Indeed, the modularity function is often used to measure the presence of community structure in networks. It is also used as a quality function in many community detection algorithms, including the widely used Louvain algorithm.

cs.SI↗

arXiv2022

Broadcasting on Paths and Cycles

math.CO↗

arXiv2022

Perfect Matchings in the Semi-random Graph Process

The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible. We focus on the problem of constructing a perfect matching in as few rounds as possible. In particular, we present an adaptive strategy for the player which achieves a perfect matching in $βn$ rounds, where the value of $β< 1.206$ is derived from a solution to some system of differential equations. This improves upon the previously best known upper bound of $(1+2/e+o(1)) \, n < 1.736 \, n$ rounds. We also improve the previously best lower bound of $(\ln 2 + o(1)) \, n > 0.693 \, n$ and show that the player cannot achieve the desired property in less than $αn$ rounds, where the value of $α> 0.932$ is derived from a solution to another system of differential equations. As a result, the gap between the upper and lower bounds is decreased roughly four times.

math.CO↗

arXiv2021

Tight Bounds on the Probabilistic Zero Forcing on Hypercubes and Grids

Zero forcing is a deterministic iterative graph colouring process in which vertices are coloured either blue or white, and in every round, any blue vertices that have a single white neighbour force these white vertices to become blue. Here we study probabilistic zero forcing, where blue vertices have a non-zero probability of forcing each white neighbour to become blue. We explore the propagation time for probabilistic zero forcing on hypercubes and grids.

math.CO↗

arXiv2020

On the existence of Hamilton cycles with a periodic pattern in a random digraph

We consider Hamilton cycles in the random digraph $D_{n,m}$ where the orientation of edges follows a pattern other than the trivial orientation in which the edges are oriented in the same direction as we traverse the cycle. We show that if the orientation forms a periodic pattern, other than the trivial pattern, then approximately half the usual $n\log n$ edges are needed to guarantee the existence of such Hamilton cycles a.a.s.

math.CO↗

arXiv2020

Localization Game for Random Graphs

We consider the localization game played on graphs in which a cop tries to determine the exact location of an invisible robber by exploiting distance probes. The corresponding graph parameter $ζ(G)$ for a given graph $G$ is called the localization number. In this paper, we improve the bounds for dense random graphs determining an asymptotic behaviour of $ζ(G)$. Moreover, we extend the argument to sparse graphs.

math.CO↗

arXiv2020

The Unit Acquisition Number of Binomial Random Graphs

Let $G$ be a graph in which each vertex initially has weight 1. In each step, the unit weight from a vertex $u$ to a neighbouring vertex $v$ can be moved, provided that the weight on $v$ is at least as large as the weight on $u$. The unit acquisition number of $G$, denoted by $a_u(G)$, is the minimum cardinality of the set of vertices with positive weight at the end of the process (over all acquisition protocols). In this paper, we investigate the Erdős-Rényi random graph process $(\mathcal{G}(n,m))_{m =0}^{N}$, where $N = {n \choose 2}$. We show that asymptotically almost surely $a_u(\mathcal{G}(n,m)) = 1$ right at the time step the random graph process creates a connected graph. Since trivially $a_u(\mathcal{G}(n,m)) \ge 2$ if the graphs is disconnected, the result holds in the strongest possible sense.

math.CO↗

arXiv2020

Hamilton Cycles in the Semi-random Graph Process

The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible. We focus on the problem of constructing a Hamilton cycle in as few rounds as possible. In particular, we present a novel strategy for the player which achieves a Hamiltonian cycle in $(2+4e^{-2}+0.07+o(1)) \, n < 2.61135 \, n$ rounds, assuming that a specific non-convex optimization problem has a negative solution (a premise we numerically support). Assuming that this technical condition holds, this improves upon the previously best known upper bound of $3 \, n$ rounds. We also show that the previously best lower bound of $(\ln 2 + \ln (1+\ln 2) + o(1)) \, n$ is not tight.

math.CO↗

arXiv2020

On Broadcasting Time in the Model of Travelling Agents

Consider the following broadcasting process run on a connected graph $G=(V,E)$. Suppose that $k \ge 2$ agents start on vertices selected from $V$ uniformly and independently at random. One of the agents has a message that she wants to communicate to the other agents. All agents perform independent random walks on $G$, with the message being passed when an agent that knows the message meets an agent that does not know the message. The broadcasting time $ξ(G,k)$ is the time it takes to spread the message to all agents. Our ultimate goal is to gain a better understanding of the broadcasting process run on real-world networks of roads of large cities that might shed some light on the behaviour of future autonomous and connected vehicles. Due to the complexity of road networks, such phenomena have to be studied using simulation in practical applications. In this paper, we study the process on the simplest scenario, i.e., the family of complete graphs, as in this case the problem is analytically tractable. We provide tight bounds for $ξ(K_n,k)$ that hold asymptotically almost surely for the whole range of the parameter $k$. These theoretical results reveal interesting relationships and, at the same time, are also helpful to understand and explain the behaviour we observe in more realistic networks.

cs.SI↗

arXiv2020

Probabilistically Faulty Searching on a Half-Line

We study $p$-Faulty Search, a variant of the classic cow-path optimization problem, where a unit speed robot searches the half-line (or $1$-ray) for a hidden item. The searcher is probabilistically faulty, and detection of the item with each visitation is an independent Bernoulli trial whose probability of success $p$ is known. The objective is to minimize the worst case expected detection time, relative to the distance of the hidden item to the origin. A variation of the same problem was first proposed by Gal in 1980. Then in 2003, Alpern and Gal [The Theory of Search Games and Rendezvous] proposed a so-called monotone solution for searching the line ($2$-rays); that is, a trajectory in which the newly searched space increases monotonically in each ray and in each iteration. Moreover, they conjectured that an optimal trajectory for the $2$-rays problem must be monotone. We disprove this conjecture when the search domain is the half-line ($1$-ray). We provide a lower bound for all monotone algorithms, which we also match with an upper bound. Our main contribution is the design and analysis of a sequence of refined search strategies, outside the family of monotone algorithms, which we call $t$-sub-monotone algorithms. Such algorithms induce performance that is strictly decreasing with $t$, and for all $p \in (0,1)$. The value of $t$ quantifies, in a certain sense, how much our algorithms deviate from being monotone, demonstrating that monotone algorithms are sub-optimal when searching the half-line.

cs.DS↗

arXiv2019

Probabilistic Zero Forcing on Random Graphs

Zero forcing is a deterministic iterative graph coloring process in which vertices are colored either blue or white, and in every round, any blue vertices that have a single white neighbor force these white vertices to become blue. Here we study probabilistic zero forcing, where blue vertices have a non-zero probability of forcing each white neighbor to become blue. We explore the propagation time for probabilistic zero forcing on the Erdős-Réyni random graph $\Gnp$ when we start with a single vertex colored blue. We show that when $p=\log^{-o(1)}n$, then with high probability it takes $(1+o(1))\log_2\log_2n$ rounds for all the vertices in $\Gnp$ to become blue, and when $\log n/n\ll p\leq \log^{-O(1)}n$, then with high probability it takes $Θ(\log(1/p))$ rounds.

math.CO↗

arXiv2019

k-regular subgraphs near the k-core threshold of a random graph

We prove that $G_{n,p=c/n}$ whp has a $k$-regular subgraph if $c$ is at least $e^{-Θ(k)}$ above the threshold for the appearance of a subgraph with minimum degree at least $k$; i.e. an non-empty $k$-core. In particular, this pins down the threshold for the appearance of a $k$-regular subgraph to a window of size $e^{-Θ(k)}$.

math.CO↗

arXiv2019

Local Clustering Coefficient of Spatial Preferential Attachment Model

In this paper, we study the clustering properties of the Spatial Preferential Attachment (SPA) model. This model naturally combines geometry and preferential attachment using the notion of spheres of influence. It was previously shown in several research papers that graphs generated by the SPA model are similar to real-world networks in many aspects. Also, this model was successfully used for several practical applications. However, the clustering properties of the SPA model were not fully analyzed. The clustering coefficient is an important characteristic of complex networks which is tightly connected with its community structure. In the current paper, we study the behaviour of C(d), which is the average local clustering coefficient for the vertices of degree d. It was empirically shown that in real-world networks C(d) usually decreases as 1/d^a for some a>0 and it was often observed that a=1. We prove that in the SPA model C(d) decreases as 1/d. Furthermore, we are also able to prove that not only the average but the individual local clustering coefficient of a vertex v of degree $d$ behaves as 1/d if d is large enough. The obtained results further confirm the suitability of the SPA model for fitting various real-world complex networks.

math.PR↗